Finite Group Databasebeta

← C3×#14219 #10632826

Subgroups — C3×#14219 #10632826

Order 768.

1542 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C211yesmaximal, characteristic
2C2×C3×#391C212yesmaximal
3C2×C3×C4×C4⋊C4C212yesmaximal
4C3×#1445C211yesmaximal, characteristic
5C3×#1441C212yesmaximal
6C3×#1436C211yesmaximal, characteristic
7C3×#1436C212yesmaximal
8C3×#1436C212yesmaximal
9C3×#1436C212yesmaximal
10#14219C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×C3×C42C2212yes—
19C3×#395C418yescomplemented
20C3×#391—28no—
21C3×C4×C4⋊C4—28no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×#395C2×C311yescharacteristic
36C2×#391C2×C312yes—
37C2×C4×C4⋊C4C2×C312yes—
38#1445C2×C311yescharacteristic
39#1441C2×C312yes—
40#1436C2×C312yes—
41#1436C2×C312yes—
42#1436C2×C312yes—
43#1436C2×C311yescharacteristic
44C22×C3×Q8—24no—
45C22×C3×Q8—24no—
46C22×C3×Q8—41no—
47C22×C3×Q8—41no—
48C22×C3×Q8C2×C414yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4C2×C418yes—
68C2×C3×C4⋊C4C2×C414yes—
69C2×C3×C4⋊C4Q818yescomplemented
70C2×C3×C42—24no—
71C2×C3×C42—24no—
72C2×C3×C42—24no—
73C2×C3×C42—24no—
74C2×C3×C42—24no—
75C2×C3×C42—24no—
76C2×C3×C42—41no—
77C2×C3×C42—42no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—41no—
86C3×#95—41no—
87C3×#95—41no—
88C3×#95C2×C418yes—
89C3×#95C2×C414yes—
90C3×#95D418yes—
91C23×Q8C22×C311yescharacteristic
92C22×C4⋊C4C22×C312yes—
93C22×C4⋊C4C22×C312yes—
94C22×C4⋊C4C22×C312yes—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C42C22×C312yes—
97C22×C42C22×C311yescharacteristic
98C22×C42C22×C312yes—
99#395C3×C418yescomplemented
100#391—28no—
101C4×C4⋊C4—28no—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C311yescharacteristic
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C311yescharacteristic
112C2×#95C22×C311yescharacteristic
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C24×C3C2411yescharacteristic
116C2×C3×Q8—216no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C2×C3×Q8—48no—
120C2×C3×Q8—44no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C22×C412yes—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C414yes—
168C22×C3×C4C22×C414yes—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×Q814yes—
173C22×C3×C4C2×Q814yes—
174C22×C3×C4C2×Q814yes—
175C22×C3×C4#4114yes—
176C22×C3×C4#4114yes—
177C22×C3×C4#4114yes—
178C3×C4⋊C4—216no—
179C3×C4⋊C4—216no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—48no—
182C3×C42—216no—
183C3×C42—44no—
184C3×C42—48no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8C2×C3×C414yes—
190C23×C4C23×C312yes—
191C23×C4C23×C311yescharacteristic
192C23×C4C23×C311yescharacteristic
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C312yes—
195C23×C4C23×C312yes—
196C23×C4C23×C312yes—
197C23×C4C23×C312yes—
198C23×C4C23×C312yes—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×C3×C418yes—
209C2×C4⋊C4C2×C3×C414yes—
210C2×C4⋊C4C3×Q818yescomplemented
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—41no—
218C2×C42—42no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95C2×C3×C418yes—
230#95C2×C3×C414yes—
231#95C3×D418yes—
232C23×C3C23×C411yescharacteristic
233C23×C3C22×D412yes—
234C23×C3C22×Q812yes—
235C23×C3C2×#4112yes—
236C23×C3C2×#4112yes—
237C23×C3C2×#4112yes—
238C23×C3H(5,𝔽2)11yescharacteristic
239C23×C3H(5,𝔽2)11yescharacteristic
240C23×C32−1+412yes—
241C3×Q8—416no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4C4×D418yes—
297C2×C3×C4C4×Q818yes—
298C2×C3×C4C22⋊Q818yes—
299C24C24×C311yescharacteristic
300C2×Q8—216no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—48no—
304C2×Q8—44no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4C22×C3×C412yes—
349C22×C4C22×C3×C412yes—
350C22×C4C22×C3×C412yes—
351C22×C4C22×C3×C414yes—
352C22×C4C22×C3×C414yes—
353C22×C4C2×C3×D414yes—
354C22×C4C2×C3×D414yes—
355C22×C4C2×C3×D414yes—
356C22×C4C2×C3×Q814yes—
357C22×C4C2×C3×Q814yes—
358C22×C4C2×C3×Q814yes—
359C22×C4C3×#4114yes—
360C22×C4C3×#4114yes—
361C22×C4C3×#4114yes—
362C4⋊C4—216no—
363C4⋊C4—216no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C42—216no—
367C42—44no—
368C42—48no—
369C22×C3C2×C4×D412yes—
370C22×C3C2×C4×Q812yes—
371C22×C3C4×#4111yescharacteristic
372C22×C3#51811yescharacteristic
373C22×C3#51911yescharacteristic
374C22×C3C2×C22⋊Q812yes—
375C22×C3#52512yes—
376C22×C3#52912yes—
377C22×C3#53312yes—
378C22×C3#53912yes—
379C22×C3#54212yes—
380C22×C3D4211yescharacteristic
381C22×C3#54612yes—
382C22×C3#54712yes—
383C22×C3D4×Q812yes—
384C22×C3#55111yescharacteristic
385C22×C3#55412yes—
386C22×C3#55511yescharacteristic
387C22×C3#55612yes—
388C22×C3#55712yes—
389C22×C3Q8211yescharacteristic
390C3×C4—28no—
391C3×C4—28no—
392C3×C4—28no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—44no—
397C3×C4—44no—
398C3×C4—82no—
399C23C23×C3×C411yescharacteristic
400C23C22×C3×D412yes—
401C23C22×C3×Q812yes—
402C23C2×C3×#4112yes—
403C23C2×C3×#4112yes—
404C23C2×C3×#4112yes—
405C23C3×H(5,𝔽2)11yescharacteristic
406C23C3×H(5,𝔽2)11yescharacteristic
407C23C3×2−1+412yes—
408Q8—416no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4C3×C4×D418yes—
464C2×C4C3×C4×Q818yes—
465C2×C4C3×C22⋊Q818yes—
466C2×C3C4×C22⋊Q812yes—
467C2×C3#233211yescharacteristic
468C2×C3#235811yescharacteristic
469C2×C3#236512yes—
470C2×C3#236911yescharacteristic
471C2×C3#244912yes—
472C2×C3#245112yes—
473C2×C3#247412yes—
474C2×C3#258812yes—
475C22C2×C3×C4×D412yes—
476C22C2×C3×C4×Q812yes—
477C22C3×C4×#4111yescharacteristic
478C22C3×#51811yescharacteristic
479C22C3×#51911yescharacteristic
480C22C2×C3×C22⋊Q812yes—
481C22C3×#52512yes—
482C22C3×#52912yes—
483C22C3×#53312yes—
484C22C3×#53912yes—
485C22C3×#54212yes—
486C22C3×D4211yescharacteristic
487C22C3×#54612yes—
488C22C3×#54712yes—
489C22C3×D4×Q812yes—
490C22C3×#55111yescharacteristic
491C22C3×#55412yes—
492C22C3×#55511yescharacteristic
493C22C3×#55612yes—
494C22C3×#55712yes—
495C22C3×Q8211yescharacteristic
496C4—28no—
497C4—28no—
498C4—28no—
499C4—44no—
500C4—44no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C3#1421911yescharacteristic, direct product factor, Sylow
506C2C3×C4×C22⋊Q812yes—
507C2C3×#233211yescharacteristic
508C2C3×#235811yescharacteristic
509C2C3×#236512yes—
510C2C3×#236911yescharacteristic
511C2C3×#244912yes—
512C2C3×#245112yes—
513C2C3×#247412yes—
514C2C3×#258812yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
10. #14219C3direct (N × H)
19. C3×#395C4semidirect
69. C2×C3×C4⋊C4Q8semidirect
99. #395C3×C4semidirect
210. C2×C4⋊C4C3×Q8semidirect
505. C3#14219direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#3952
C2×C3×#3912
C2×C3×C4×C4⋊C42
C3×#14452
C3×#14412
C3×#14362
#142193